21 research outputs found
The asymptotic spectrum of graphs and the Shannon capacity
We introduce the asymptotic spectrum of graphs and apply the theory of
asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new
dual characterisation of the Shannon capacity of graphs. Elements in the
asymptotic spectrum of graphs include the Lov\'asz theta number, the fractional
clique cover number, the complement of the fractional orthogonal rank and the
fractional Haemers bounds
The asymptotic spectrum of LOCC transformations
We study exact, non-deterministic conversion of multipartite pure quantum
states into one-another via local operations and classical communication (LOCC)
and asymptotic entanglement transformation under such channels. In particular,
we consider the maximal number of copies of any given target state that can be
extracted exactly from many copies of any given initial state as a function of
the exponential decay in success probability, known as the converese error
exponent. We give a formula for the optimal rate presented as an infimum over
the asymptotic spectrum of LOCC conversion. A full understanding of exact
asymptotic extraction rates between pure states in the converse regime thus
depends on a full understanding of this spectrum. We present a characterisation
of spectral points and use it to describe the spectrum in the bipartite case.
This leads to a full description of the spectrum and thus an explicit formula
for the asymptotic extraction rate between pure bipartite states, given a
converse error exponent. This extends the result on entanglement concentration
in [Hayashi et al, 2003], where the target state is fixed as the Bell state. In
the limit of vanishing converse error exponent the rate formula provides an
upper bound on the exact asymptotic extraction rate between two states, when
the probability of success goes to 1. In the bipartite case we prove that this
bound holds with equality.Comment: v1: 21 pages v2: 21 pages, Minor corrections v3: 17 pages, Minor
corrections, new reference added, parts of Section 5 and the Appendix
removed, the omitted material can be found in an extended form in
arXiv:1808.0515
The asymptotic induced matching number of hypergraphs: balanced binary strings
We compute the asymptotic induced matching number of the -partite
-uniform hypergraphs whose edges are the -bit strings of Hamming weight
, for any large enough even number . Our lower bound relies on the
higher-order extension of the well-known Coppersmith-Winograd method from
algebraic complexity theory, which was proven by Christandl, Vrana and Zuiddam.
Our result is motivated by the study of the power of this method as well as of
the power of the Strassen support functionals (which provide upper bounds on
the asymptotic induced matching number), and the connections to questions in
tensor theory, quantum information theory and theoretical computer science.
Phrased in the language of tensors, as a direct consequence of our result, we
determine the asymptotic subrank of any tensor with support given by the
aforementioned hypergraphs. In the context of quantum information theory, our
result amounts to an asymptotically optimal -party stochastic local
operations and classical communication (slocc) protocol for the problem of
distilling GHZ-type entanglement from a subfamily of Dicke-type entanglement
Towards a Geometric Approach to Strassen's Asymptotic Rank Conjecture
We make a first geometric study of three varieties in (for each ), including the Zariski
closure of the set of tight tensors, the tensors with continuous regular
symmetry. Our motivation is to develop a geometric framework for Strassen's
Asymptotic Rank Conjecture that the asymptotic rank of any tight tensor is
minimal. In particular, we determine the dimension of the set of tight tensors.
We prove that this dimension equals the dimension of the set of oblique
tensors, a less restrictive class introduced by Strassen.Comment: Final version. Revisions in Section 1 and Section
Barriers for fast matrix multiplication from irreversibility
Determining the asymptotic algebraic complexity of matrix multiplication,
succinctly represented by the matrix multiplication exponent , is a
central problem in algebraic complexity theory. The best upper bounds on
, leading to the state-of-the-art , have been
obtained via the laser method of Strassen and its generalization by Coppersmith
and Winograd. Recent barrier results show limitations for these and related
approaches to improve the upper bound on .
We introduce a new and more general barrier, providing stronger limitations
than in previous work. Concretely, we introduce the notion of "irreversibility"
of a tensor and we prove (in some precise sense) that any approach that uses an
irreversible tensor in an intermediate step (e.g., as a starting tensor in the
laser method) cannot give . In quantitative terms, we prove that
the best upper bound achievable is lower bounded by two times the
irreversibility of the intermediate tensor. The quantum functionals and
Strassen support functionals give (so far, the best) lower bounds on
irreversibility. We provide lower bounds on the irreversibility of key
intermediate tensors, including the small and big Coppersmith--Winograd
tensors, that improve limitations shown in previous work. Finally, we discuss
barriers on the group-theoretic approach in terms of "monomial"
irreversibility